In more recent years, there has been a shift to learned and potentially non-convex priors [1]}, [2]}, [3]}, [4]}, [5]}. It has been observed that these methods often outperform classical methods. Moreover, a full convergence analysis has been be provided [5]}, [2]}. However, such an analysis assumes minimizers of the Tikhonov functional to be known or at least be given within a certain accuracy. For non-convex regularizers such an assumption is unrealistic and global minimizers are challenging to compute. Instead, when trying to find a regularized solution one often employs minimization algorithms such as gradient descent or variations thereof which converge to critical points (such as local minimizers close to the initial guess) rather than to global minimizers of the Tikhonov functional. While one could constrain the learned regularizers to only include convex functionals [8]}, [4]} this might result in suboptimal reconstructions when the underlying signal class is inherently non-convex. For such classes non-convexity of the regularizer can be a highly desirable property and as such a convergence analysis for this case is needed. Importantly, such an analysis should not rely on the strict assumption that the regularized solutions are global minimizers of the underlying Tikhonov functional.
